What's the difference between poker with public cards and without them?
It depends a little on what you mean by "the same rounds, just with no public cards dealt."

If you mean that each player will just be dealt 2 cards, and no public cards exist, then really we're playing a sort of "high card" game. The best hand is just a pair of aces, CFR will solve this quickly, because the number of possible game states is extremely small compared to a full poker game (especially if we exploit the symmetry of suits, since flushes aren't possible).

If you mean that each player will be dealt 5 cards, with several rounds of betting as before, CFR will probably do less well. The state space will be larger, since there are more cards in play (10 instead of 9). Betting may become more complex, and more complex betting expands the state space enormously. 

If you mean that the cards are dealt as before, but the program simply will not look at the cards, then you've kept the state space of the game the same size, but radically reduced the number of information sets. Playing against an opponent who can look at the cards on the table, your program would be at an enormous disadvantage. For instance, imagine the cards on the table are "3 3 8 8 5", and that you have a pair of 2's in your hand. You would want to play very differently from if the cards on the table were "2 2 10 4 7", but an AI without access to the table cards would have to act the same in both situations.